Symplectomorphism group of $T^{\ast }(G_{\mathbb{C}}\mathrm{/}B)$ and the braid group I: a homotopy equivalence for $G_{\mathbb{C}}=S{L}_3(\mathbb{C})$

For a semisimple Lie group $G_{\mathbb{C}}$ over $\mathbb{C}$, we study the homotopy type of the symplectomorphism group of the cotangent bundle of the flag variety and its relation to the braid group. We prove a homotopy equivalence between the two groups in the case of $G_{\mathbb{C}}=S{L}_3(\mathbb{C})$, under the $SU(3)$-equivariancy condition on symplectomorphisms.